view 'homotopy addition lemma and corollary
|
diff
|
2009-05-02 14:13:35
- revision [
Version = 13
-->
Version 14
]
by
bci1
|
\subsection{Homotopy addition lemma}
\emph{Let $f: \boldsymbol{\rho}^\square(X) \to \mathsf D$ be a morphism of
\PMlinkname{double groupoids}{HomotopyDoubleGroupoidOfAHausdorffSpace}
with connection. If $\alpha \in {\boldsymbol{\rho}^\square_2}(X)$ is thin, then $f(\alpha)$
is thin.}
\subsubsection{Remarks}
The groupoid ${\boldsymbol{\rho}^\square_2}(X)$ employed here is as defined by the
\PMlinkname{cubically thin homotopy}{CubicallyThinHomotopy} on the set
$R^{\square}_2(X)$ of \PMlinkname{squares}{ThinDoubleTracks}. Additional explanations of the data, including concepts such as path groupoid and homotopy double groupoid are provided in an
\PMlinkname{attachment}{WeakHomotopyAdditionLemma}.
\subsection{Corollary}
\emph{Let $u : I^3\to X$ be a singular cube in a Hausdorff space $X$.
Then by restricting $u$ to the faces of $I^3$ and taking the
corresponding elements in $\boldsymbol{\rho}^{\square}_2 (X)$, we obtain a
cube in $\boldsymbol{\rho}^{\square} (X)$ which is commutative by the Homotopy
addition lemma for $\boldsymbol{\rho}^{\square} (X)$ (\cite{BHKP}, Proposition
5.5). Consequently, if $f : \boldsymbol{\rho}^{\square} (X)\to \mathsf{D}$ is
a morphism of
\PMlinkname{double groupoids}{HomotopyDoubleGroupoidOfAHausdorffSpace} with connections, any singular cube
in $X$ determines a
\PMlinkexternal{commutative {3-shell}}{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in $\mathsf{D}$.}
\begin{thebibliography}{9}
\bibitem{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space, {\it Theory and Applications of Categories.} \textbf{10},(2002): 71-93.
\end{thebibliography}
|
|
|
